Residual and Past Entropy for Concomitants of Ordered Random Variables of Morgenstern Family

For a system, which is observed at time t, the residual and past entropies measure the uncertainty about the remaining and the past life of the distribution, respectively. In this paper, we have presented the residual and past entropy of Morgenstern family based on the concomitants of the different types of generalized order statistics (gos) and give the linear transformation of such model. Characterization results for these dynamic entropies for concomitants of ordered random variables have been considered.


Introduction
The Morgenstern family discussed in [1] provides a flexible family that can be used in such contexts, which is specified by the distribution function (df) and the probability density function (pdf), respectively, as follows: , (, ) =   ()   () [1 +  (2  () − 1) (2  () − 1)] , where −1 ≤  ≤ 1, and   (),   () and   (),   () are the marginal pdf and df of  and , respectively.The parameter  is known as the dependence parameter of the random variables  and .If  is zero, then  and  are independent.
The concept of gos was introduced by Kamps [3] and we refer to it as case-I of gos; Kamps and Cramer [4] have introduced a second model of gos which we refer to as case-II of gos.The concept of lower gos was given by Pawlas and Szynal [5], and later Burkschat et al. [6] introduced it as dual generalized order statistics (dgos) to enable a common approach to descendingly ordered random variables like reversed order statistics and lower records models.
Di Crescenzo and Longobardi [12] have introduced past entropy over (0, ), since it is reasonable to presume that in many realistic situations uncertainty is not necessarily related to the future but can also refer to the past.They have also shown the necessity of past entropy and its relation with the residual entropy.If  denotes the lifetime of an item or of a living organism, then past entropy (or uncertainty of lifetime distribution) of an item is defined as where   () is the reversed hazard rate of  given by   ()/  ().In this paper, we will obtain and study the residual and past entropy of the Morgenstern family for concomitants of ordered random variables.We will also consider the characterization results based on the entropy function for concomitants of ordered random variables based on residual lifetime distribution and the past life distribution.The organization of the paper is as follows.In Section 2, we obtain the distribution function for concomitants of ordered random variables of the Morgenstern family.In Section 3, we obtain the residual and past entropy of our model and study the linear transformation and the upper bound for concomitants of ordered random variables.Some characterization results are presented in Section 4. Finally, some conclusions and comments are given in Section 5.

Distribution Function for Concomitants of Ordered Random Variables
Let (  ,   ),  = 1, 2, . . ., , be  pairs of independent continuous nonnegative random variables from some bivariate population with df (, ).From (1), the conditional distribution function of  given  =  is given by For the Morgenstern family with conditional distribution function given by ( 11), the df of the concomitant of case-I of gos  [,,,] , 1 ≤  ≤ , is given by where  (,,,) () is the pdf of case-I of gos  [,,,] (see Kamps [3]), the df of the concomitant of case-I of dgos  [,,,] , 1 ≤  ≤ , is given by where  (,,,) () is the pdf of case-I of dgos  [,,,] (see Burkschat et al. [6]), the df of the concomitant of case-II of gos  [,, m,] , 1 ≤  ≤ , is given by where  (,, m,) () is the pdf of case-II of gos  [,, m,] (see Kamps and Cramer [4]), and the df of the concomitant of case-II of dgos  [,, m,] , 1 ≤  ≤ , is given by where  (,, m,) () is the pdf of case-II of dgos  [,, m,] (see Kamps and Cramer [4] and Burkschat et al. [6]).Equations ( 4) to ( 7) and ( 12) to ( 15) can be combined as follows: where In the following section we will use the last equations to obtain and study the residual and past entropy of concomitants for Morgenstern family based on the types of gos.

Theorem 1. For any absolutely continuous random variable 𝑌 *
, which is the concomitant of th ordered random variable of Morgenstern family defined in (18).From (9) where From ( 16), (17), and (10), the past entropy for concomitants of ordered random variables of Morgenstern family is given by the following theorem.

Theorem 2. For any absolutely continuous random variable 𝑌 *
, which is the concomitant of th ordered random variable of Morgenstern family defined in (18).From (10) where  2 (, , , , , ) The following proposition gives the values of the functions ( with equality in which when  → 0, ( *  ) is the Shannon entropy based on the concomitant of ordered random variables.

Some Characterization Results
Gupta et al. [13] have studied characterizations of residual and past entropy of order statistics.Here, in this section, we present some characterization results based on residual and past entropy of the concomitant of ordered random variables.
Consider a problem of finding sufficient condition for the uniqueness of the solution of the initial value problem (IVP): where  is a given function of two variables whose domain is a region  ⊂ R 2 , ( 0 ,  0 ) is a specified point in , and  is the unknown function.By the solution of the IVP on an interval  ⊂ R, we mean a function () such that (i)  is differentiable on , (ii) the growth of  lies in , (iii) ( 0 ) =  0 , and (iv)   () = (, ( 0 )), for all  ∈ .The following theorem together with other results will help in proving our characterization result.
Proof (see Gupta and Kirmani [14]).For any function (, ) of two variables defined in  ⊂ R 2 , we now present a sufficient condition which guarantees that the Lipschitz condition is satisfied in .
Lemma 5. Suppose that the function  is continuous in a convex region  ⊂ R 2 .Suppose further that / exists and is continuous in .The function  satisfies Lipschitz condition in .
Proof (see Gupta and Kirmani [14]).We now present our two characterization results.Next we present a characterization result of the linear mean residual family of the concomitant of th ordered random variable based on Theorem 6.The linear mean residual life of the concomitant of th ordered random variable is given by    () =  + ,  > 0,  > −1.It can be verified that the corresponding failure rate is given by   *  () = (1+)/(+ ).The linear mean residual family of a distribution has been studied among others by Hall and Wellner [15], Oakes and Dasu [16], and Gupta and Kirmani [14].It includes the exponential distribution for  = 0 and the power distribution for −1 <  < 0. Gupta et al. [13] have studied the linear mean residual life of a distribution based on order statistics.

Conclusion and Comments
The different types of gos are considered in one model based on the concomitants for Morgenstern family.From this model, we derived an analytical expression of the residual and past lifetime distribution and find some formulas of it.The dynamic entropy measures based on the concomitant of ordered random variables characterize uniquely the underlying distribution and are also bounded above in terms of Shannon entropy of the concomitant of the different types of gos.Stochastic comparisons can also be studied easily for such model.

Theorem 4 .
Let the function  be defined and continuous in a domain  ⊂ R 2 , and let  satisfy a Lipschitz condition (with respect to ) in ; namely,      (,  1 ) −  (,  2 )     ≤       1 −  2     ,  > 0, is the reversed hazard rate of the concomitant of th ordered random variable, for  = 1, 2.